Reconstruction of depth-4 multilinear circuits with top fan-in 2

Ankit Gupta*, Neeraj Kayal, Satya Lokam

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


We present a randomized algorithm for reconstructing multilinear ΣΠΣΠ(2) circuits, i.e., multilinear depth-4 circuits with fan-in 2 at the top + gate. The algorithm is given blackbox access to a polynomial f ε double-struck F[x 1,...,x n] computable by a multilinear ΣΠΣΠ(2) circuit of size s and outputs an equivalent multilinear ΣΠΣΠ(2) circuit, runs in time poly(n,s), and works over any field double-struck F. This is the first reconstruction result for any model of depth-4 arithmetic circuits. Prior to our work, reconstruction results for bounded depth circuits were known only for depth-2 arithmetic circuits (Klivans & Spielman, STOC 2001), ΣΠΣ(2) circuits (depth-3 arithmetic circuits with top fan-in 2) (Shpilka, STOC 2007), and ΣΠΣ(k) with k = O(1) (Karnin & Shpilka, CCC 2009). Moreover, the running times of these algorithms have a polynomial dependence on |F| and hence do not work for infinite fields such as Q. Our techniques are quite different from the previous ones for depth-3 reconstruction and rely on a polynomial operator introduced by Karnin et al. (STOC 2010) and Saraf & Volkovich (STOC 2011) for devising blackbox identity tests for multilinear ΣΠΣΠ(k) circuits. Some other ingredients of our algorithm include the classical multivariate blackbox factoring algorithm by Kaltofen & Trager (FOCS 1988) and an average-case algorithm for reconstructing ΣΠΣ(2) circuits by Kayal.

Original languageEnglish
Title of host publicationSTOC '12 - Proceedings of the 2012 ACM Symposium on Theory of Computing
Number of pages17
StatePublished - 2012
Externally publishedYes
Event44th Annual ACM Symposium on Theory of Computing, STOC '12 - New York, NY, United States
Duration: 19 May 201222 May 2012

Publication series

NameProceedings of the Annual ACM Symposium on Theory of Computing
ISSN (Print)0737-8017


Conference44th Annual ACM Symposium on Theory of Computing, STOC '12
Country/TerritoryUnited States
CityNew York, NY


  • algebraic complexity
  • circuit reconstruction


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